Abstract

Reduced-order models (ROMs) have been proven to be a useful tool for efficient simulations of the responses of large-scale dynamical systems and their identification. Here I focus on ROM ’s applications to the solution of the nonlinear inverse coefficient problems for linear PDEs. Our framework circumvents this nonlinearity by introducing a family of recursive nonlinear data preprocessing procedures, relying on sparse network realizations of the data-driven (aka noninvasive) ROMs. This procedure “absorbs” much of the nonlinearity of the problem, thus making the subsequent imaging or inversion a lot more straightforward. The uniqueness of Calderón formulations was proven by several authors (the speaker included) starting from the early 1980s, however applicability of the network approximations in this setting became only understood after works of de Verdiere, Curtis, Ingerman and Morrow in the 1990s. I begin with the 1D inverse Sturm-Liouville problem and estimate its electrical conductivity by embedding its network approximation. Then I outline generalizations of this approach for 2D Calderón formulations with complete and partial DtN data using planar graphs and explain intrinsic difficulties for extensions to dimensions >2 due to curse of dimensionality. Finally I present a ROM based Lippmann-Schwinger inversion algorithm for wave problems and consider its application to the synthetic aperture radar (SAR) problem in a multiple-scattering environment, which avoids the curse of dimensionality. The efficiency of this approach was recently improved with the help of a novel data-completion algorithm, allowing to lift  SAR (monostatic) data set to the multi-input/multi-output  one. Liliana Borcea, Fernando Guevara Vasquez, David Ingerman, Leonid Knizhnerman, Alexander Mamonov, Shari Moskow, Andy Thaler, Mikhail Zaslavskiy and Jörn Zimmerling contributed to different stages of this research.